Projective Geometry

What is Projective Geometry?

In Euclidean geometry can be described with lines and circles, its tolls being the straight-edge and the compass. In Euclidean geometry there is also a measure of distance and compare figures by measuring them. In projective geometry we throw out the compass, leaving only the straight-edge. We no longer measure anything and relate sets of points by projectivity.

Some Definitions

A range is the set of all points on a lin.

A pencil is the set of all lines that go through a point.

Axioms of Projective Geometry

Axiom 1: There exist a point and a line that are not incident.

Axiom 2: Every line is incident with at least three distinct points.

Axiom 3: Any 2 distinct points are incident with just one line.

Axiom 4: If A, B, C, D are four distinct points such that lines AB meets CD then AC meets BD.

Axiom 5: If ABC is a plane, there is at least one point not in the plane ABC.

Axiom 6: Any two distinct planes have at least two common points.

Axiom 7: The three diagonal points of a complete quadrangle are never collinear.

Axiom 8: If a projectivity leaves invariant each of three distinct points on a line, it leaves invariant every point on the line.

These eight axioms govern projective geometry. Before looking at the concept of duality in projective geometry, let's look at a few theorems that result from these axioms.

A Few Theorems

Theorem If two lines have a common point, they are coplanar.

Lets say C is our common point, then let the lines be AC and BC. Thus they line in the plane ABC.

Theorem Any two distinct lines have at most one common point.

Assume that two given lines have two common point A and B. Axiom 3 tells us thatany two distinct points are incident with just one line, so these two given lines must be the same line, thus leading to a contradiction.

Duality in Projective Geometry

The principle of duality states that we can interchange the words point and line, and every denifition is still relevant and theorems remain true. This also follow that terminology like join and meet, vertex and side, and so on.

Since this is the case let's look at the two previous basic theorems that were stated and interchange the words line and point.

Theorem If two points have a common line, they are coplanar.

This is obvious since two points always have a common line. Since they both fall on the same line they must be coplanar.

Theorem Any two distinct point have at most one common line.

Once again this is can easily be proven, since this statement is reflected in Axiom 3.

Hence one can see that if the dual statement is true so is it's primal. One can then has a choice, solve the primal or the dual. Leaving someone with the option of what representation they are most comfortable with.

The same can be said for the principle of duality in 3-dimensions, where points, lines, and planes are interchanged with planes, lines, and points.